Resonant impurity states in semiconductors in a strong magnetic field

We present a theory of magnetically induced resonance states in narrow-gap semiconductors and apply it to study the resonance states in InSb. By expanding the impurity wave functions in terms of the free-electron Landau wave functions, we obtain an infinite set of coupled equations. If the magnetic field is sufficiently large, i.e., $\gamma >1\mathrm{}$ (here, $\gamma =\hslash {\omega }_{\frac{c}{2}}{\mathrm{Ry}}^{*}$), only a small number of Landau levels need be solved self-consistently because for any energy to a good approximation, the coupling need be considered only between the adjacent Landau level and all the lower ones. The calculation of the energy and width can be made using a multicomponent generalization of the Kohn variational method for phase shifts. We have made detailed calculations for the lowest resonant state associated with the $n=1\mathrm{}$ Landau level. Screened potentials were used primarily because they simplify the numberical calculation, but the procedure is applicable without modification to any potential $V\left(z\right)\mathrm{}$ which goes to zero faster than $\frac{1}{z}$. Furthermore, although we have used the parabolic band model, the method can be readily modified to include nonparabolicity.